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Lesson 7.1
Angles of Polygons
Essential Question:
How can I find the sum of the
measures of the interior angles of a
polygon?
Polygon
• A plane figure made of three or more
segments (sides).
• Each side intersects exactly two other sides at
their endpoints.
• Polygons are named by vertices in
consecutive order, going CW or CCW.
Diagonals in a Polygon
• A segment that joins two non-consecutive vertices.
• The diagonals from one vertex divide a polygon into
triangles.
Interior Angle Sum of a Triangle
4 3 5
m1 = m4
180
1
m2 = m5
2
m4
m1 + m5
m2 + m3 = 180
1   180
7.1 Angles of Polygons
January 18, 2017
Polygon Interior Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
7.1 Angles of Polygons
Sum of
Interior
Angles
180
January 18, 2017
Interiore Angle Sum in a Quadrilateral
2 s  360
2
3
1
4
6
m1 + m2 + m3 = 180
5
m4 + m5 + m6 = 180
m1 + m4 + m2 + m5 + m3 + m6 = 360
7.1 Angles of Polygons
January 18, 2017
Polygon Interor Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
180
Quadrilateral
4
2
360
7.1 Angles of Polygons
Sum of
Interior
Angles
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Interior Angle Sum in a Pentagon
From this vertex, how many
diagonals are there?
2
7.1 Angles of Polygons
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Angle Sum in a Pentagon
How many triangles
are there?
3
And what is the sum
of the angles of
each triangle?
180
180
180
180
7.1 Angles of Polygons
January 18, 2017
Angle Sum in a Pentagon
3 s  540
So what is the sum of
the interior angles of
a pentagon?
180
180
180
3  180 = 540
7.1 Angles of Polygons
January 18, 2017
Polygon Interior Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
180
Quadrilateral
4
2
360
Pentagon
5
3
540
7.1 Angles of Polygons
Sum of
Interior
Angles
January 18, 2017
Interior Angle Sum in a Hexagon
How many diagonals
from this vertex?
3
How many triangles
are formed?
4
The sum of the angles
is?
4  180 = 720
7.1 Angles of Polygons
4 s  720
January 18, 2017
Polygon Interior Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
180
Quadrilateral
4
2
360
Pentagon
5
3
540
Hexagon
6
4
720
7.1 Angles of Polygons
Sum of
Interior
Angles
January 18, 2017
7.1 Angles of Polygons
January 18, 2017
Polygon Interior Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
180
Quadrilateral
4
2
360
Pentagon
5
3
540
Hexagon
6
4
720
Octagon
8
6
1080
7.1 Angles of Polygons
Sum of
Interior
Angles
January 18, 2017
What’s the pattern?
• A polygon with n sides can be divided into
how many triangles?
•n – 2
• The sum of the angles then is?
• 180(n – 2)
7.1 Angles of Polygons
January 18, 2017
Polygon Interior Angle Sums
Polygon
Sides
Triangles
formed by
Diagonals
Triangle
3
1
180
Quadrilateral
4
2
360
Pentagon
5
3
540
Hexagon
6
4
720
Octagon
8
6
1080
n-gon
n
n–2
180(n – 2)
7.1 Angles of Polygons
Sum of
Interior
Angles
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Theorem 7.1 Polygon Interior Angles Theorem
The sum of the interior angles of a convex ngon is:
180(n  2)
memorize this!
7.1 Angles of Polygons
January 18, 2017
Example 1:
Find the sum of the interior angles of a
polygon with 14 sides.
180(14 – 2) = 180(12)
= 2160
7.1 Angles of Polygons
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Example 2:
The sum of the interior angles of a polygon is
2700. How many sides does the polygon
have? 17
180(n – 2) = 2700
180n – 360 = 2700
180n = 3060
n = 17
7.1 Angles of Polygons
January 18, 2017
Example 3:
The sum of the interior angles of a polygon
is 1620. How many sides does the polygon
have? 11
180(n – 2) = 1620
180n – 360 = 1620
180n = 1980
n = 11
7.1 Angles of Polygons
January 18, 2017
Example 4:
The sum of the interior angles of a polygon
is 1380. How many sides does the polygon
have? This is not a polygon.
180(n – 2) = 1380
180n – 360 = 1380
180n = 1740
Why must this be a whole number?
7.1 Angles of Polygons
n = 9.66666…
January 18, 2017
7.1 Corollary to the Polygon Interior
Angles Theorem
The sum of the interior angles of a quadrilateral is
360.
4
1
2
3
1 + 2 + 3 + 4 = 360
Example 5:
Solve for x.
x
55
x
x + x + 55 + 55 = 360
2x + 110 = 360
2x = 250
x = 125 °
Your Turn
Find the value of x in the diagram.
x° + 108° + 121° + 59° = 360°
x + 288 = 360
x = 72 °
Regular Polygons
Regular Polygon
• All sides congruent
• All angles congruent
• The Sum of the interior angles is 180(n – 2)
• Since the angles are congruent, the measure of
each interior angle in a regular polygon is
180°(𝑛 − 2)
𝐸𝐼 =
𝑛
January 18, 2017
7.1 Angles of Polygons
Example 6:
Find the measure of each angle of a regular
pentagon. 108
180(5  2) 180(3)

5
5
108
540

 108
108
108
5
7.1 Angles of Polygons
108
108
January 18, 2017
Example 7:
Each angle of a regular polygon measures
160. How many sides does the polygon
have?
180(n  2)
 160
n
180n  360  160n
20n  360
n  18
January 18, 2017
7.1 Angles of Polygons
Example 8:
A home plate for a baseball field is shown.
a. Is the polygon regular? Explain your reasoning.
The polygon is not equilateral or equiangular.
So, the polygon is not regular.
Example 8:
b. Find the measures of ∠C and ∠E.
180° ⋅ (n − 2) = 180° ⋅ (5 − 2) = 540°
x° + x° + 90° + 90° + 90° = 540°
2x + 270 = 540
2x = 270
x = 135
Therefore, ∠C= 135° and ∠E= 135°
The Sum of the Exterior Angles
This always means using one exterior angle
at each vertex.
But not
this one.
or this angle.
This angle…
7.1 Angles of Polygons
January 18, 2017
The Sum of the Exterior Angles
Extend only ONE side at each vertex.
Exterior
Angle
Exterior
Angle
Exterior
Angle
7.1 Angles of Polygons
January 18, 2017
Theorem 7.2 Polygon Exterior Angles
Theorem
The sum of the exterior angles of any polygon,
one angle at each vertex, is 360.
𝑆𝐸 = 360°
m∠1 + m∠2 + · · · + m∠n = 360°
Example 9:
Find the value of x in the diagram.
x° + 2x° + 89° + 67° = 360°
3x + 156 = 360
3x = 204
x = 68 °
Corollary
The measure of an exterior
angle of a regular polygon
with n sides is
360°
𝐸𝐸 =
𝑛
7.1 Angles of Polygons
January 18, 2017
Example 10:
Find the measure of an exterior
angle of a regular 40-gon.
Solution:
360/40 = 9
7.1 Angles of Polygons
January 18, 2017
Example 15:
The trampoline shown is shaped like a regular
dodecagon.
a. Find the measure of each interior angle.
180(n  2) 180(12 − 2)
=
12
n
180(10)
=
12
1800
= 150°
=
12
Example 15:
The trampoline shown is shaped like a regular
dodecagon.
b. Find the measure of each exterior angle.
360
= 30°
12
Summary
• The sum of the interior angles of an n-gon is 180(n – 2).
• The sum of the exterior angles of any polygon is 360.
• The measure of an interior angle of a regular polygon is
180(𝑛−2)
.
𝑛
360
• The measure of an exterior angle of a regular polygon is
.
𝑛
7.1 Angles of Polygons
January 18, 2017